trend-based analysis of a population model of the akap sca

Trend-based Analysis of a Population Model ofthe AKAP Scaffold Protein

Oana Andrei and Muffy Calder

School of Computing Science, University of Glasgow, G12 8RZ, UK,[email protected],[email protected]

Abstract. We formalise a continuous-time Markov chain with multi-dimensional discrete state space model of the AKAP scaffold protein asa crosstalk mediator between two biochemical signalling pathways. Theanalysis by temporal properties of the AKAP model requires reasoningabout whether the counts of individuals of the same type (species) areincreasing or decreasing. For this purpose we propose the concept ofstochastic trends based on formulating the probabilities of transitionsthat increase (resp. decrease) the counts of individuals of the same type,and express these probabilities as formulae such that the state spaceof the model is not altered. We define a number of stochastic trendformulae (e.g. weakly increasing, strictly increasing, weakly decreasing,etc.) and use them to extend the set of state formulae of ContinuousStochastic Logic. We show how stochastic trends can be implemented ina guarded-command style specification language for transition systems.We illustrate the application of stochastic trends with numerous smallexamples and then we analyse the AKAP model in order to characteriseand show causality and pulsating behaviours in this biochemical system.

1 Introduction

In the recent years biochemical networks have become an important applicationarea for modelling approaches and analysis techniques developed in theoret-ical computer science. Our approach to modelling and analysing biochemicalnetworks is stochastic processes, continuous-time Markov chains (CTMCs) inparticular, which allow new quantitative analysis in addition to the traditionalsimulation afforded by ordinary differential equations (ODEs). CTMC modelswhere states represent the counts of molecules for each biochemical species, alsocalled molecular CMTCs, together with the behaviour analysis based on Gille-spie’s stochastic simulation algorithm [1], provide a faithful representation ofbiochemical networks. One major limitation of the molecular CTMCs is the sizeof the underlying state space that can easily become too large to be handled ex-plicitly by stochastic model checking tools. CTMCs with levels [2] are based ondiscrete levels of concentration instead of exact molecule counts. In comparisonto the molecular CTMC, the level abstraction reduces the state space, leading tomodels that are more amenable to model checking techniques that analyse the

2 Andrei and Calder

entire state space. Another limitation of the molecular CTMCs is the need forprecise molecular concentrations for the species and details about the reactions,whereas CTMCs with levels allow for greater abstraction and relative quantities.

We focus here on modelling the scaffold protein AKAP and its role as a medi-ator of the crosstalk between the cyclic AMP (cAMP) and the Raf-1/MEK/ERKsignalling pathways. The behaviour of this biochemical system is complex andstill under study in the laboratory. Following discussions with laboratory sci-entists, we have developed a CTMC with levels model, which we believe to bethe first formal model of the system. This modelling paradigm is well-suited tothe AKAP system because the experimental data gathered so far are relativerather than exact. In other words, exact rates of reactions are unknown, buttheir relative rates are known; for example, some are known to be about threetimes faster than others, etc. Typical questions and properties conjectured bylaboratory scientists include if increasing concentration levels of molecule A leadto decreasing concentration levels of molecules B and C, or confirming the pul-sating behaviour suggested by the lab experiments. In order to formalise theseconjectured properties in the AKAP model we define stochastic trends.

Stochastic trends stem from modelling biochemical networks but they can bemore generally applied to Markov Population Processes (MPPs) – continuous-time Markov chains where states record the counts of individuals in each colonyof a population [3–6]. MPPs can be used for modelling in a wide variety of appli-cation domains, including, for example, computer networks, chemical reactionsnetworks, and ecology networks. Birth-death processes are simple MPPs. In par-ticular, molecular CMTCs and CTMCs with levels are examples of MPPs. Manykey questions to ask of Markov population models involve trends. For example,is a particular colony increasing/decreasing, is the change strict, weak, etc., orif we get more individuals in colony A, will colony B then decrease? Analysisof such logical properties by model checking requires a suitable representationof trends. We propose an approach based on formulating the probabilities oftransitions that increase (resp. decrease) colony counts in a stochastic model.

Related work. The concept of a trend in a discrete or continuous deterministicsetting is well established (e.g. slope or first-order derivative), but less so in astochastic setting. First-order derivatives have been considered previously in thecontext of model checking biochemical systems. For example in BIOCHAM [7,8], oscillatory properties are analysed using queries expressed as formulae in LTLwith constraints over real numbers. Such formulae are interpreted over traces ofstates and a state includes not only the concentration value of each molecularspecies but also the value of its first-order derivative. This analysis applies toBIOCHAM deterministic semantics, where the underlying model has exactlyone trace and therefore the concept of a trend is encapsulated by the first-order derivative. In this paper we consider the concept of trend in a stochasticsetting, and without explicitly storing the trend in a state variable. Oscillatingbehaviours can be formulated either as temporal formulae [9–11] in CTL, PCTLor CSL or based on a system of differential equations [12]. However, for theAKAP model we have to deal with incomplete data about the reaction rates.

Trend-based Analysis of a Population Model of the AKAP Scaffold Protein 3

Stochastic trends provide a preliminary analysis technique when only partialinformation is provided on the reaction rates such as a reaction rate is of theorder of some other reaction rate. Trend formulae are very closely related tothe trend variable approach [13]. One advantage of trend formulae over trendvariables is that the use of trend formulae does not increase the size of thestate space. Moreover, our analysis is forward-looking, or a priori, based on theprobability (over all possible transitions) for a colony to increase (resp. decrease),whereas trend variables imply an a posteriori analysis based on behaviour thathas already occurred. In Sect. 5.3 we will give an in depth comparison betweentrend formulae and trend variables.

Contributions. This paper is an extension of previous work [14] and focuseson introducing stochastic trends as an analysis technique for MPPs in general,models of biochemical networks in particular. The contributions of the paper aretwofold:

– Stochastic trend formulae for characterising the probability of increasing ordecreasing colonies that can be used to extend the set of state formulae intemporal logics such as Continuous Stochastic Logic, along with an encodingof trend formulae in the guarded-command modelling language of the PRISMprobabilistic model checker.

– A CTMC with level model of the AKAP scaffold protein as a mediator ofthe crosstalk between the cyclic AMP and the Raf-1/MEK/ERK signallingpathways, and the use of stochastic trends to characterise causality andpulsating behaviours in the AKAP model.

Outline. The next section reviews the definition and basic concepts of continuous-time Markov chains (CTMCs), Markov population processes (MPPs) and CTMCswith levels. We also review the reagent-centric modelling style of MPPs, biochem-ical systems in particular, and their representation in the modelling language ofthe PRISM model checker, and the temporal logic Continuous Stochastic Logicfor expressing properties about CTMCs in PRISM. Section 3 presents the biolog-ical model of the AKAP scaffold protein and in Sect. 4 we define the associatedCTMC with level model. We introduce stochastic trends in Sect. 5 and usethem for analysing the behaviour of the AKAP system in Sect. 6. We give ourconclusions and directions for future work in Sect. 7.

2 Preliminaries

In the following, we assume some familiarity with continuous-time Markov chains,see for example [15–17].

2.1 Continuous-time Markov Chains

Definition 2.1. A (labelled) continuous-time Markov chain (CTMC) is a tu-ple (S, s0, R, L) where S is a countable set of states, s0 ∈ S the initial state,

4 Andrei and Calder

R : S × S → R≥0 the transition rate matrix, AP a finite set of atomic proposi-tions, and L : S → 2AP the labelling function associating to each state in S theset of atomic propositions from AP that are valid in that state.

The transition rates determine the probability of transitions to be completedwithin a certain amount of time following the negative exponential distribution:when R(s, s′) > 0, then the probability of this transition to be triggered within ttime units equals 1− e−R(s,s′)·t. The time spent in state s before any transitionis triggered is exponentially distributed with parameter:

E(s) =∑s′∈S

R(s, s′).

E(s) is called the exit rate of state s. For a given state s, there is a race be-tween outgoing transitions from s if there are more than one state s′ such thatR(s, s′) > 0. If the exit rate of a state is equal to 0 then no transition can be firedfrom it and the state is called absorbing. The time-abstract probability of a states′ to be the next state to which a transition is made from state s is computedby a transition probability function P : S × S → [0, 1] as follows:

P(s, s′) =

R(s,s′)E(s) if E(s) 6= 0

1 if E(s) = 0 and s = s′

0 otherwise

This transition probability function, together with the state space S, initialstate s0 and labelling function L define a discrete-time Markov chain (DTMC)embedded in the CTMC.

An infinite path of a CTMC is a sequence s0t0s1t1 . . . such thatR(si, si+1) > 0and ti ∈ R>0 denotes the time spent in state si for all i ≥ 0. A finite path is asequence s0t0s1t1 . . . sk−1tk−1sk such that sk is an absorbing state. A self-looptransition is a single transition going back to the same state it fired from. Acycle is a path beginning and ending with the same state.

2.2 Markov Population Processes

A population is a collection of individuals grouped into colonies or categoriesbased on common features. Markovian population processes (MPPs) [3–6] arecontinuous-time Markov chains that express demographic processes such as birthand immigration (addition of individuals), death (removal of individuals) oremigration (transfer of individuals between colonies). The characteristic featureof MPPs is given by their states which enumerate the counts of individuals inevery colony.

Definition 2.2. A Markov population process (MPP) is a continuous-timeMarkov chain M = (S, s0, R,AP , L) with S defined as a set of n-dimensionalstates of the form s = (x1, . . . , xn) with n ≥ 1 the number of colonies in thepopulation and xi a non-negative integer representing the number of individualsin colony i, for all i, 1 ≤ i ≤ n.


// WVUTPQRS0

λ ,, WVUTPQRS1

λ ,,

µll . . .

λ ,,

µll ONMLHIJKi

λ --

µll WVUTPQRSi+ 1

λ ,,

µll . . .

µmm

Fig. 1. An M/M/1 queue with initial state 0, user arrival rate λ and user departurerate µ.

We define an MPP either component-wise according to Def. 2.2, through astate-transition graph if the state space is relatively small, or via a set of reactionsif we model a biochemical network. The graphical state-transition notation weuse for an MPP is the usual one for CTMCs: a directed graph with states asnodes and an edge between any pair of nodes si and sj if R(si, sj) > 0 withR(si, sj) the edge label; the initial state is marked by an incoming arrow withno source.

Example 2.1. The simplest example of an MPP is a birth-death process (BD pro-cess) defined as an MPP with one colony. In a BD process states can be indexedby non-negative integers representing the counts of individuals in the single-colony population such that state transitions occur only between neighbouringstates: from i to i+ 1 or from i+ 1 to i. One straightforward application of BDprocesses is in queueing theory. A BD process is an example of a single serverqueue with an infinite buffer size – also known as the M/M/1 queue in Kendall’snotation [18, 19] if the user arrival (birth) rate λ and user departure (death) rateµ are independent of the colony size. Then each state represents the number ofusers in the system. In Fig. 1 we depict the MPP model of such a queue.

Another application domain for MPPs is biochemical networks. In these net-works, the species quantities are usually given in terms of concentrations. Givena biochemical network represented as a set of reactions and initial concentra-tions for each species, we can associate an MPP model with as many coloniesas the number of different species and where both chemical species and reactionrates are expressed in terms of number of molecules, assuming that all chemicalspecies are in the same static compartment (i.e. of constant volume V ). Thistype of MPP model is usually referred to as molecular CTMC because we countthe molecules. We translate a concentration c for a species X to a number ofmolecules equal to C = c · V ·NA where NA is Avogadro’s number (the numberof molecules contained in a mole of X).

A reaction is usually given by a stoichiometric equation:

r : α1X1 + . . .+ αnXnk−→ β1X1 + . . .+ βnXn (1)

where, for all i, 1 ≤ i ≤ n, non-negative integers αi and βi are the stoichio-metric coefficients defining how many molecules of Xi are consumed and pro-duced respectively by the reaction, k is the constant reaction rate coefficient.The species on the left and right hand side with non-zero coefficients are calledreactants and products respectively. In practise we do not include species withnull stoichiometric coefficients in a stoichiometric equation. Let Xmax

i denote

6 Andrei and Calder

the upper bound on the number of molecules Xi. Such bounds can either beobtained from experimental data or estimated by using stochastic simulationand model checking in tandem [20], or simply imposed for any species thatgrows infinitely (in order to guarantee a finite CTMC). The reaction can occurfrom a state s if s − (α1, . . . , αn) ≥ 0 and s − (α1, . . . , αn) + (β1, . . . , βn) ≤(Xmax

1 , . . . , Xmaxn ), where Xmax

i is the maximum possible number of moleculesXi. If a transition from s is taken according to this reaction, then we move tothe state s′ = s−(α1, . . . , αn)+(β1, . . . , βn). Assuming mass-action kinetics, thetransition rate is proportional to the number of affected molecules and equalsk ·∏

1≤i≤n(Ci

αi

), with Ci denoting the number of molecules of species Xi, since

we need to consider all possible combinations of individual molecules.The combinatorics of every possible molecular count in a molecular CTMC

can lead to state space explosion. Molecular CTMC models can be too large toanalyse using model checking and only an analysis based on stochastic simula-tion becomes available, which does not construct the complete underlying statespace. One way of tackling this problem is to discretise each species concentra-tion uniformly into a number of levels of concentration, rather than representingby numbers of molecules. A transition from one state to another reflects changesof these levels according to a biochemical reaction. The result is a stochastic,population based model that is more abstract than the molecular CTMC andcalled continuous-time Markov chain with levels [21, 22, 2]. One advantage ofusing CTMC with levels is that it allows one to deal with incomplete or onlyrelative information about molecular concentrations, often the case in experi-mental settings. Another advantage of CTMCs with levels over the molecularCTMC is its smaller state space, allowing the models to be more amenable tostochastic model checking.

Informally, in a CTMC with levels each species is characterised by a numberof levels, equidistant from each other, with step size h. We assume that all specieshave the same step size. We assign to each species different concentration levels,from 0 (corresponding to null concentration) to a maximum number N . Whenthe maximum molar concentration is M , then the step size h = M

N . Here, weassume all reactions have mass-action kinetics.

Definition 2.3 (CTMC with levels). A CTMC with levels for a biochemicalsystem is an MPP where the molecules of the same species form a colony andstates represent levels of concentrations of the species. For n different species(Xi)1≤i≤n, a state is a tuple s = (`1, `2, . . . , `n) with `i the discrete concentrationlevel for the species Xi, for all i, 1 ≤ i ≤ n. A reaction of the form given byEq. 1 has similar firing conditions as in the case of molecular CTMCs and therate of a transition fired by such a reaction is the product of the reaction ratecoefficient adjusted for the step-size discretisation h and the concentrations ofthe reacting species, i.e., k

h ·(`α11 ·h) · . . . ·(`αn

n ·h), where k is the constant reactionrate coefficient and `αi

i is the discrete level of concentration of reacting speciesXi with stoichiometric coefficient αi.

In comparison with molecular CTMCs and traditional ordinary differentialequations (ODEs), CTMCs with levels models are more compact than molecular


CMTCs yet retain the stochasticity lost in the ODEs. The granularity of aCTMC with levels can be changed by decreasing the stepsize. As the stepsizedecreases with the number of levels tending to infinity, the variability of theCTMC with levels model is reduced and, as predicted by Kurtz’s Theorem [23](on the relationship of the class of density dependent Markov chains and a setof ODEs), the obtained global behaviour of the CTMC with levels model tendstowards that given by the ODE model [2].

A biochemical reaction does not take only the simple form of an arrival,departure or transfer event between colonies as the definition of MPP. Oftenthere is a form of cooperative transfer from some colonies to others. An exampleis the following where species are transferred to and from X1 and X2, and X3.

Example 2.2. Consider a simple reaction system consisting of three species X1,X2 and X3 with initial molar concentrations X1(0) = X2(0) = 2 mol/l and

X3(0) = 0 mol/l , and a forward and a backward reaction X1 + X2k1−→ X3,

X3k2−→ X1 + X2 with k1 = 1.2, k2 = 0.2. If we consider N = 3 the maximum

number of levels of concentration, the step size is h = 23 mol/l . The CTMC with

levels modelling this system is represented in Fig. 2 with the initial state rep-

resenting the initial concentration levels given by (bX1(0)h c, bX2(0)

h c, bX3(0)h )c =

(3, 3, 0). We convert a molar concentration Xi(0) to a number of levels bXi(0)h c.

//�� (3, 3, 0)

16.2 --�� (2, 2, 1)

7.2 --

0.3

mm

�� (1, 1, 2)

1.8 --

0.6

mm

�� (0, 0, 3)

0.9

mm

Fig. 2. CTMC with levels for the forward/backward reaction X1 +X2k1/k2←→ X3

2.3 Modelling MPPs in PRISM

There are several languages or formalisms for specifying Markov population pro-cesses based on rate transition matrix descriptions, state-transition graphs andstoichiometric equations for chemical reactions. Several other formalisms areavailable for this purpose as overviewed in [6], including guarded command mod-els (GCM). GCMs are textual models describing the classes of possible statetransitions on colonies and take inspiration from Dijkstra’s guarded-commandlanguage (GCL). Reactive Modules [24] and PRISM’s specification language [25]are based on the same formalism.

We adopt the reagent-centric modelling approach to modelling biochemicalsystems [26] implemented as a PRISM specification as follows. Each of thecolonies, also called reagents, in an interaction or transition is mapped to aprocess, whose variation reflects increase or decrease, e.g., through productionor consumption, through birth, death or migration, etc. For example, the chem-

ical reaction r1 given in stoichiometric notation by X1 + X2k1−→ X3 refers to

8 Andrei and Calder

three reagents and so it is modelled by three processes, X1, X2 and X3, whichare then composed concurrently, synchronising on the event r1. If we assume anunderlying semantics of CTMC with levels, after the event r1, the concentrationlevel of X3 is increased and those of X1 and X2 are decreased.

The PRISM language includes modules with local variables, action-labelledguarded commands (transitions) and multiway synchronisation of modules. Eachprocess is implemented by a module, and the modules are composed using themultiway synchronisation operator (denoted by ||) over all common actions.

We illustrate the approach with the biochemical system introduced in Ex-ample 2.2. The PRISM model, depicted in Fig. 3, has three modules X1, X2 andX3, one for each species, all modules running concurrently. Each module has theform: a state variable denoting the species concentration level, followed by com-mands labelled by the reactions in which the species is a reactant or product.In this example there are two commands labelled by r1 and r2. Each commandhas the form:

[label] guard -> rate : update;

meaning that the module makes a transition to a state described by the updateat the given rate when the guard is true (the label is optional). The r1-labelledcommand in the first two modules decreases the number of levels by 1 and in thethird module increases the number of levels by 1. Initially, there are N levels ofX1 and X2 and 0 levels of X3. The module Const consists of commands labelledby the reaction labels with trivial guards and updates and the rate equal to theconstant reaction rate coefficient. All r1-labelled transitions synchronise andthe resulting transition occurs with a rate equal to the product of the individualrates, i.e. (k1/h)*(X1*h)*(X2*h).

2.4 Stochastic Model Checking

Since MPPs are CTMCs, we use Continuous Stochastic Logic (CSL) [15] as atemporal logic for specifying properties about their stochastic behaviour. CSL isa stochastic extension of the Computational Tree Logic (CTL) allowing one toexpress a probability measure of the satisfaction of a temporal property in eithertransient or in steady-state behaviours. The formulae of CSL are state formulaeand their syntax is the following:

State formula Φ ::= true | a | ¬Φ | Φ ∧ Φ | P./ p[Ψ ] | S./ p[Ψ ]

Path formula Ψ ::= XΦ | ΦUI Φ

where a ranges over a set of atomic propositions AP , ./∈ {≤, <,≥, >}, p ∈ [0, 1],and I is an interval of R≥0. There are two types of CSL properties: transient (ofthe form P./ p[Ψ ]) and steady-state (of the form S./ p[Ψ ]). A formula P./ p[Ψ ] istrue in state s, denoted by s |= P./ p[Ψ ], if the probability that Ψ is satisfied bythe paths starting from state s meets the bound ./ p. A formula S./ p[Ψ ] is truein a state s if the steady-state (long-run) probability of being in a state whichsatisfies Ψ meets the bound ./ p. The path formulae are constructed using the


ctmc

const double max_conc = 2;

const int N = 4;

const double h = max_conc/N;

const double k1 = 1.2;

const double k2 = 0.2;

module X1

X1 : [0..N] init N;

[r1] (X1>0) -> (X1*h) : (X1’=X1-1);

[r2] (X1<N) -> (1) : (X1’=X1+1);

endmodule

module X2

X2 : [0..N] init N;

[r1] (X2>0) -> (X2*h) : (X2’=X2-1);

[r2] (X2<N) -> (1) : (X2’=X2+1);

endmodule

module X3

X3 : [0..N] init 0;

[r1] (X3<N) -> (1) : (X3’=X3+1);

[r2] (X3>0) -> (X3*h) : (X3’=X3-1);

endmodule

module Const

[r1] true -> (k1/h) : true;

[r2] true -> (k2/h) : true;

endmodule

system

X1 || X2 || X3 || Const

endsystem

Fig. 3. PRISM program for the forward/backward reaction X1 +X2k1/k2←→ X3

X (next) operator and the UI (time-bounded until) operator. Informally, thepath formula XΦ is true on a path starting in s if Φ is satisfied in the next

state following s in the path, whereas Φ1 UI Φ2 is true on a path ω if Φ2 holdsat some time instant in the interval I in a state s′ in ω and at all preceding timeinstants Φ1 holds. This is a minimal set of operators for CSL. The operators false,disjunction and implication can be derived using basic logical equivalences. Twomore path operators are available as syntactic sugar:

– the eventually operator F (future) where FI Φ ≡ true UI Φ, and

– the always operator G (globally) where GI Φ ≡ ¬(FI ¬Φ).

If I = [0,∞), then the temporal operators U, F, G are no longer time-bounded,hence we omit the interval superscript notation in this situation.

The model checking problem of a state formula Φ being satisfied in an MPPis denoted by M, s0 |= Φ. We omit the initial state s0 when it is obvious.

The PRISM probabilistic model checker [17] has a property specificationlanguage based on the temporal logics PCTL, CSL, LTL and PCTL∗, includingextensions for quantitative specifications and rewards. PRISM allows one toexpress a probability measure that a temporal formula is satisfied. The bound ./pmay not be specified, in which case a probability is calculated in PRISM. Thusthese two additional properties P=?[Ψ ] and S=?[Ψ ] are available: the results of theverification of such formulae are the expected probabilities for the satisfactionof the path formula denoted by Ψ .

10 Andrei and Calder

3 The AKAP Scaffold Protein

In this section we give an overview of the AKAP scaffold protein and its medi-ating role in the crosstalk between the cAMP and Raf-1/ERK/MEK signallingpathways. The behaviour of this system is complex and still under study in thelaboratory.

In intracellular signal transduction pathways, scaffolds are proteins exhibitingtwo main functions [27]. Namely, a scaffold protein anchors particular proteinsin specific intracellular locations for receiving signals or transmitting them, andit provides a catalytic function by increasing the output of a signalling cascadeor decreasing the response time for a faster output under certain circumstances.

3.1 Species

Figure 4 illustrates the species involved in the biochemical system and their in-teractions in the AKAP model with emphasis on the AKAP’s anchoring role aspositions on the scaffold are filled or unfilled. The species involved are: cyclicadenosine monophosphate (cAMP); protein kinase A (PKA); Raf-1 with twophosphorylation sites of interest, Serine 338 (Ser338) and Serine 259 (Ser259);phosphodiesterase 8 (PDE8A1); phosphatase PP. The left-hand side of Fig. 4shows an unfilled scaffold with free PDE8A1, and the right-hand side shows afilled scaffold.

S259PP Raf-1

PKA

PDE8A1

cAMP

AKAP

S388

S259PP

PKA

PDE8A1

S388

Raf-1

cAMP

AKAP

Fig. 4. Interactions between cAMP, unfilled AKAP scaffold, free PDE8A1 and filledscaffold where: A→ B means A activates or phosphorylates B, A 99K B stands for Adephosphorylates B, A a B means A degrades B.


3.2 Behaviour

If the concentration of cAMP rises above a given threshold, cAMP activates PKAby binding to it. Activated PKA catalyses the transfer of phosphates to the phos-phorylation site Ser259 of Raf-1. The site Ser338 of Raf-1 is said to be inhibitedwhen Ser259 is phosphorylated. Only when Ser338 becomes phosphorylated, thepathway Raf-1/MEK/ERK is activated (and say that Raf-1 is active) and thesignalling cascade begins.

The catalytic function of PKA sometimes couples with the AKAP, by bindingPKA together with phosphodiesterase PDE8A1 on the scaffold to form a complexthat functions as a signal module. Under these conditions, as the cell is stim-ulated, cAMP activates PKA, and then PKA is responsible for the activation ofPDE8A1 (by phosphorylation). PDE8A1 degrades cAMP, but if phosphorylated,PDE8A1 degrades more cAMP, hence rapidly reducing the amount of cAMP thatcan activate PKA. This leads to a feedback mechanism for downregulating PKA.

Discussions with laboratory scientists revealed the following expectations, orconjectures, about the AKAP system behaviour.

Causal relation between concentration fluctuations. We define causality to mean:assuming more A (less A) denotes increasing (resp. decreasing) concentrationlevels for a species A, the implication “more A⇒ less B” means that a decreasein B’s concentration level is necessarily preceded by an increase in A’s concen-tration level. Laboratory scientists expect that increasing concentration level ofphosphorylated PDE8A1 leads to a cascade of changes in the concentration lev-els of the other reactants: decreasing concentration levels of cAMP and activePKA, and an increase in the activity of Raf-1 – due to lower levels of phospho-rylated Raf-1 at site Ser259. Informally, we express this causality relation by thefollowing relationship:

more pPDE8A1 ⇒ less cAMP ⇒ less active PKA ⇒ more active Raf-1

Pulsating behaviour. Time courses from laboratory experiments suggest the pres-ence of a pulsating behaviour in the system. The pulsations ensure that the stateof the Raf-1 pathways alternates between active and inactive, which is a desirablebehaviour because very long periods of activity or inactivity may increase therisk of disease. In the current model we do not consider explicitly interactionsbetween cAMP and Raf-1. However, the system is not closed and we include anexogenous interaction represented by the diffusion of cAMP. We conjecture thismakes the system exhibit a pulsating behaviour corresponding to the feedbackmechanism for the downregulation of PKA, coupled with the diffusion of cAMP.Note that we call such a behaviour pulsating, not oscillating: oscillation assumesfluctuation around a given value, but the current partial data do not provide uswith such a value.


4 MPP Model for the AKAP System

We define a CTMC with levels model for the AKAP system based on combina-tions of the species represented in Fig. 4. An overview of the model follows.

4.1 Scaffolded Species

The AKAP scaffold has three positions to be filled in order by PKA, site Ser259of Raf-1 and PDE8A1 respectively, with the third one not necessarily filled. Wedefine an abstraction over these species consisting of different combinations inorder to encode the context of reactions as follows:

– for filled scaffold: S[PKA’s state][Ser259’s state][PDE8A1’s state]

– for unfilled scaffold: S[PKA’s state][Ser259’s state].

where each state has a binary representation with 1 representing activated orphosphorylated and 0 otherwise. For instance, S100 represents a filled AKAPscaffold with active PKA and unphosphorylated Ser259 and PDE8A1, whereasS01 represents an unfilled scaffold with inactive PKA and phosphorylated Ser259.All the possible abstract species involving a scaffold are: S00, S10, S01, S11, S000,S100, S101, S110, S011, S010, S001, S111.

We also distinguish between PDE8A1 and its phosphorylated form PDE8A1(pPDE8A1) as two different unscaffolded species. The same reasoning applies tothe phosphatase PP anchored on a filled scaffold and PP on an unfilled scaffold(denoted by uPP). The remaining unscaffolded species is cAMP.

4.2 Biochemical Reactions

In Fig. 5 we list the biochemical reactions of the model. Each reaction is givenby a stoichiometric equation with explicit reference to the scaffold positions(the underlying reactions have mass-action kinetics). We associate reaction rateconstants (from r1 to r26) with each biochemical reaction.

The existing experimental data suggest only approximate ratios of the reac-tion rates. More precisely, we have some information on the ratio between therate of PKA phosphorylating Raf-1 at site Ser259 and PDE8A1 (either on thescaffold or not). On unfilled scaffolds, PKA phosphorylates three times less un-scaffolded PDE8A1 than Raf-1 at site Ser259 from the same scaffold. On filledscaffolds, PKA phosphorylates Raf-1 at Ser259 and PDE8A1 at the same rate.Consequently the relation between constant rates of the reactions involving PKAphosphorylating either PDE8A1 or Raf-1 is: r4 = r5 = r6 = r10 = r11 =3 · r12 = 3 · r13. In addition, phosphorylated PDE8A1 degrades about threetimes more cAMP than PDE8A1 does, hence we deduce the following ratiosbetween the constants rates of the reactions where PDE8A1 degrades cAMP :r19 = r20 = r21 = r22 = r23 = r24 = 3 · r25 = 9 · r26.


cAMP diffusion:r1→ cAMP

PKA activation:

S000 + cAMPr2→ S100

S00 + cAMPr3→ S10

Ser259 phosphorylation:

S100r4→ S110

S101r5→ S111

S10r6→ S11

Ser259 dephosphorylation:

PP + S010r7→ PP + S000

PP + S011r8→ PP + S001

PP + S01r9→ PP + S00

cAMP release:

S111r17→ S011 + cAMP

S11r18→ S01 + cAMP

PDE8A1 phosphorylation:

S100r10→ S101

S110r11→ S111

S10 + PDE8A1r12→ S10 + pPDE8A1

S11 + PDE8A1r13→ S11 + pPDE8A1

PDE8A1 dephosphorylation:

PP + S001r14→ PP + S000

PP + S011r15→ PP + S010

uPP + pPDE8A1r16→ PP + PDE8A1

cAMP degradation:

S011 + cAMPr19→ S011

S001 + cAMPr20→ S001

S100 + cAMPr21→ S100

S110 + cAMPr22→ S110

S010 + cAMPr23→ S010

S000 + cAMPr24→ S000

pPDE8A1 + cAMPr25→ pPDE8A1

PDE8A1 + cAMPr26→ PDE8A1

Fig. 5. Biochemical reactions occurring during scaffold-mediated crosstalk betweencAMP and the Raf-1/MEK/ERK pathway. The notation Sv1v2v3 represents a filled scaf-fold with v1, v2, v3 denoting the activation state of the bound PKA, site Ser259 ofRaf-1 and PDE8A1 respectively, i.e., 0 for inactive and 1 for active or phosphorylated.Similarly, Su1u2 represents an unfilled scaffold with u1 and u2 denoting the activationstate of the bound PKA and Ser259 respectively.

4.3 The PRISM Model for the AKAP System

The PRISM model consists of four modules: one module for cAMP, one modulefor the scaffold with 12 variables (one variable for each type of scaffold), a modulefor PDE8A1 and pPDE8A1, and a module for PP and uPP. The complete PRISMmodel can be found at http://www.dcs.gla.ac.uk/~muffy/akap/.

We assume that the initial concentrations for species S00, S000, PP and uPPare all equal to 12mol/l, for cAMP 120mol/l, for unscaffolded PDE8A1 6mol/l,and 0 otherwise. We calculate the stepsize for the CTMC with levels abstractionas h = 12

N withN the number of levels. The system is not closed as cAMP is addedexogenously from time to time. Such interaction is needed in the model becausecAMP is consumed and to avoid termination, must be replenished. We modelthis interaction with an extra integer variable tick ranging from 0 to maximumvalue tick_max (10 in our prototype). The concentration level of cAMP increaseswhen the value of tick is less than tick_max/2 or it reaches the maximum value


(with tick being reset to 0). The variable tick is incremented by 1 whether ornot diffusion takes place, i.e. when its value is greater than tick_max/2 but lessthan the maximum. We consider a default 1.0 rate for all reactions including r4and r19, unless defined as equal or proportional to r4 and r19.

An indication of the size of the model is: for N = 3, we have 1 632 240 statesand 12 691 360 transitions, and for N = 5, we have 74 612 328 states and 734259 344 transitions.

5 Trend-based Characterisation of Transitions in an MPP

In this section we define trend formulae that describe stochastic trends of colonies,illustrate them with several examples, show how they can be encoded in thePRISM model checker, and compare them to the trend variables introduced byBallarini and Guerriero [13].

5.1 Trend Formulae

In a similar approach taken to the definition of the transition probability func-tion, we introduce families of functions Pi↑, Pi↓, Pi= corresponding to increas-ing, decreasing or constant counts of individuals in a colony i respectively, wherei ranges over colony identifiers in an MPP.

Definition 5.1. LetM = (S, s0, R,AP , L) be an MPP. The probability of mak-ing a transition from a state s to a state where the count of individuals in colonyi increases is a function Pi↑ : S → [0, 1] defined as the sum of all i-increasingtransition rates divided by the exit rate in state s:

Pi↑(s) =

{1

E(s) ·∑{R(s, s′) | s′ ∈ S, si < s′i}, if E(s) 6= 0

0, otherwise

The functions Pi↓ : S → [0, 1] and Pi= : S → [0, 1] of making a transition froma state s to a state where the count of individuals in colony i decreases or staysconstant are defined in a similar way:

Pi↓(s) =

{1

E(s) ·∑{R(s, s′) | s′ ∈ S, si > s′i}, if E(s) 6= 0

0, otherwise

Pi=(s) =

{1

E(s) ·∑{R(s, s′) | s′ ∈ S, si = s′i}, if E(s) 6= 0

0, otherwise

As expected, we have Pi↑(s)+Pi↓(s)+Pi=(s) = 1 for all s ∈ S with E(s) 6= 0.

Definition 5.2 (Trend formulae). A trend formula θ is a boolean predicateover Pi↑(s), Pi↓(s) and Pi=(s), where s ∈ S, of one of the following forms:

θ ::= f(s) = p | f(s) > p | f(s) = f ′(s) | f(s) > f ′(s) | ¬ θ | θ ∧ θ∀f, f ′ ∈ {Pi↑,Pi↓,Pi=},∀s ∈ S,∀p ∈ [0, 1]


Using the above elementary trend formulae, we define a derived set of trendformulae consisting of inequalities such as Pi↑ ≤ p or Pi↑ ≥ Pi↓ and the follow-ing auxiliary named trends.

Definition 5.3 (Auxiliary trend formulae). We say that in a state s thestochastic trend of a colony i is:

– strictly increasing if

i⇑(s) , (Pi↑(s) > Pi↓(s)) ∧ (Pi↑(s) > Pi=(s)) is true

– strictly decreasing if

i⇓(s) , (Pi↓(s) > Pi↑(s)) ∧ (Pi↓(s) > Pi=(s)) is true

– weakly increasing if i↑(s) , Pi↑(s) > Pi↓(s) is true

– weakly decreasing if i↓(s) , Pi↓(s) > Pi↑(s) is true

– very weakly increasing if i↑= (s) , Pi↑(s) ≥ Pi↓(s) is true

– very weakly decreasing if i↓= (s) , Pi↓(s) ≥ Pi↑(s) is true

– constant if i=(s) , (Pi=(s) > Pi↓(s)) ∧ (Pi=(s) > Pi↑(s)) is true

– equi if il= (s) , (Pi↑(s) = Pi↓(s)) ∧ (Pi↓(s) = Pi=(s)) is true

We illustrate the use of trend formulae in the next section.

5.2 Trend-based Properties in CSL

We use trend formulae in CSL formulae for reasoning over changes in particularcolony counts. Therefore, we extend the set of state formulae in CSL to includetrend formulae as modalities of arity 0. The definition of path formulae does notchange.

State formula Φ ::= true | a | Φ ∧ Φ | θ | P./ p[Ψ ] | S./ p[Ψ ]

Path formula Ψ ::= XΦ | ΦUI Φ

The semantics of trend formulae is defined as s |= θ if and only if θ(s) ≡ true.In the following we illustrate several CSL properties using trend formulae on

two examples.

Example 5.1. Consider MPP C1 defined in Fig. 6 with one colony whose countranges from 1 to 5 individuals and states from s0 to s6. The initial state iss0 = 4. We encode the MPP using two variables: i for the colony index and k forthe state index. Then for instance, the evaluation of the transition probabilityfunctions Pi↑, Pi↓ and Pi= in state s0 gives 0, 1 and 0 respectively (hence i⇑(s0)is true), while in state s1, functions Pi↑, Pi↓ and Pi= evaluate to 1

2 , 12 and 0

respectively. In state s2 the trend of i is strictly increasing since Pi↑(s2) = 1. Instate s4 the probability of a decreasing count is 2

3 and of a constant count is 13 .

In the following CSL experiments we use the variable s to range over states(indexed by k), with k ranging from 0 to 6:


Fig. 6. Markov population process C1 and C2 both having initial state s0.

– The probabilities of reaching a state sk where the trend θ is true, with θranging over i⇑, i↑, i↑=, i⇓, i↓, i↓=, i= and il=, are listed in Table 1.

– Eventually all states are stochastically very weakly increasing:C1 |= P≥1[F (P>0[G (i↑=)])] returns true.

– Not all states are eventually weakly increasing:C1 |= P≤0[F (P>0[G (i↑)])] returns true.

– Eventually all states are stochastically very weakly decreasing:C1 |= P≥1[F (P>0[G (i↓=)])] returns true.

– The probability that all states are very weakly decreasing:C1 |= P=?[G (i↓=)] returns 0.5.

Table 1. Model checking CSL formulae for C1 that compute the probability of reachinga state sk where the trend θ is true, with θ ranging over trend formulae and k rangingfrom 0 to 6

Model checking CSL formula k

0 1 2 3 4 5 6

C1 |= P=?[F ((i⇑) ∧ (s = k))] 0 0 0.5 0 0 0 0

C1 |= P=?[F ((i↑) ∧ (s = k))] 0 0 0.5 0 0 0 0

C1 |= P=?[F ((i↑=) ∧ (s = k))] 0 1 0.5 0.5 0 0 0.5

C1 |= P=?[F ((i⇓) ∧ (s = k))] 1 0 0 0 0.5 0.167 0

C1 |= P=?[F ((i↓) ∧ (s = k))] 1 0 0 0 0.5 0.167 0

C1 |= P=?[F ((i↓=) ∧ (s = k))] 1 1 0 0.5 0.5 0.167 0.5

C1 |= P=?[F ((i=) ∧ (s = k))] 0 0 0 0.5 0 0 0.5

C1 |= P=?[F ((il=) ∧ (s = k))] 0 0 0 0 0 0 0


– Eventually i will strictly decrease for some time with a non-zero probabilityuntil it reaches a constant trend:C1 |= P≥1[F (P>0[(i⇓) U (i=)])] returns true.

– Eventually i will strictly increase for some time with a non-zero probabilityuntil it reaches a constant trend:C1 |= P≥1[F (P>0[(i⇑) U (i=)])] returns true.

Example 5.2. Consider now the MPP C2 defined in Fig. 6. Note that C2 hasinfinite paths including infinite loop-free paths (i.e., infinite paths without self-loops). We analyse a set of CSL queries using trends that are more complex thanthose from Example 5.1. Again, k ranges from 0 to 6.

– Eventually a state with an equi trend is reached (more precisely s1):C2 |= P≥1[F (il=)] returns true, whereas in C1 this query returns false.

– The probability of reaching a state having a very weakly increasing trend andin the next state the trend is strictly decreasing with non-zero probability:C2 |= P=?[F ((i↑=) ∧ P>0[X ((i⇓) ∧ (s = k))])] returns 1 for k = 4, 0.5 fork = 5, and 0 otherwise.If we restrict the probability of a strictly decreasing next state to at least0.5, then the probability of C2 |= P=?[F ((i↑=) ∧ P>0.5[X ((i⇓) ∧ (s = k))])]is 0.5 for k = 5, and 0 otherwise.

– The probability that eventually all states are stochastically very weakly de-creasing: C2 |= P=?[FP>0[G (i ↓=)]] returns 0.5, whereas in C1 the samequery returns 1.

– The probability that eventually i will strictly decrease for some finite timewith a non-zero probability until it reaches a constant trend:C2 |= P=?[FP>0[(i⇑) U (i=)]] returns 0.5.

– Eventually i will have a strictly decreasing trend for some time until reachinga constant trend and then, with probability greater than 0.5, will show anincreasing trend:C2 |= P≥1[FP>0[(i⇓) UP>0[(i=) UP>0.5[i⇑]]]] returns true.

– The probability that always a decreasing trend of i eventually leads to anincreasing trend and vice versa:C2 |= P=?[G (((i⇓) =⇒ P>0[F i⇑]) ∧ ((i⇑) =⇒ P>0[F i⇓]))] returns 0.5.

– The long-run probability that a decreasing trend of i eventually leads to anincreasing trend and vice versa:C2 |= S≥1[((i⇓) =⇒ P>0[F i⇑]) ∧ ((i⇑) =⇒ P>0[F i⇓])] returns true.

– Variable i has a constant trend in the long-run, more specifically when states3 is reached: C2 |= S=?[i

=] returns 0.5.

Since we can define i-increasing/decreasing/constant functions for DTMCs,the trend formulae approach presented in this section is also applicable to DTMCmodels and PCTL formulae.


5.3 Trend Formulae vs. Trend Variables

An approach closely related to trend formulae is described in [13] in the contextof modelling and analysis of biochemical systems. It is based on associating twoboolean variables inc X and dec X to each species X in order to record, foreach possible transition, if the value of X increases or decreases respectively; ifthe variable X is not updated by a transition, neither are the associated vari-ables inc X and dec X. The aim is to analyse behavioural queries such as mono-tonic and oscillatory trends in models of biochemical systems. In our preliminarywork [28] we had a similar approach based on adding one integer variable drv Xto each species X; the value of drv X is updated at the the same time as thevalue of X and it denotes the sign of the change of X: 1 for increasing, -1 fordecreasing and 0 otherwise. In the following we identify two major differencesbetween the trend variable approach and our trend formulae approach.

State Space Size. Trend formulae do not increase the size of the state space, awell-known issue for the trend variable approach. To support this claim let usfirst give a constructive definition of a single-colony MPP enriched with trendvariables. Let M = (S, s0, R, L) be a single-colony MPP; then the correspondingMPP with trend variables M ′ = (S′, s′0, R

′, L′) is constructed as follows:

1. Add the initial state s′0 = (s0, t, t) to S′.2. For all states i, j ∈ S with R(i, j) > 0 and (i, inc, dec) ∈ S′ with inc, dec ∈{t, f}, add (j, inc′, dec′) to S′ where:

inc′ =

t, if i < j

f , if i > j

inc, if i = j

dec′ =

f , if i < j

t, if i > j

dec, if i = j

3. R((i, inc, dec), (j, inc′, dec′)) = R(i, j) for all (i, inc, dec), (j, inc′, dec′) ∈ S′.4. L′(i, inc, dec) = L(i) for all (i, inc, dec) ∈ S′.

Proposition 5.1. Given a single-colony MPP M , the state size of the MPP M ′

obtained from M by enriching it with trend variables is greater or at least equalto the state size of M .

Proof. We prove that |S′| ≥ |S′| by identifying two types of transitions in Mthat increase the state space:

– If R(i, j) > 0 for i > j and (i, inc, dec) ∈ S′, then (j, f , t) ∈ S′. If R(k, j) > 0for k < j and (k, inc′, dec′) ∈ S′, then (j, t, f) ∈ S′. In this case M ′ has twodistinct states for the same colony count of j, one more than M has.

– If R(i, j) > 0 and R(j, i) > 0 with i > j and (i, inc, dec), (j, inc′, dec′) ∈ S′,then (j, f , t), (i, t, f) ∈ S′. If (inc, dec) 6= (t, f) then M ′ has two distinctstates for the same colony count of i, one more than M has; the same rea-soning goes for the state j in M .


Fig. 7. Markov population process C′2 with initial state s′0 = (4, t, t) obtained from C2

by enriching the states with two boolean variables keeping track of the increasing ordecreasing trend of the first component of the state.

Consider the following simple example. We add trend variables to the MPPC2 defined in Fig. 6 to obtain the MPP C ′2 depicted in Fig. 7. Notice that C ′2has one additional state and one additional transition, due to the cycle betweenstates s5 and s6 in C2: if we consider a path in C ′2 starting from the initial state(4, t, t), when we first reach the state where i = 1 the trend variables inc anddec are set to f and t respectively because the value of i decreases strictly; butwhen the state i = 1 (i.e., state s′7) is the state visited from i = 0 (i.e., state s′6),then inc and dec are set to t and f respectively since the value of i increasesfrom 0 to 1. Hence in C ′2 there are two states with i = 1 but different values forthe trend variables inc and dec.

The result above can be generalised for MPPs with several colonies. Thereforeif a state occurs multiple times along an execution path or along different paths,the size of the state space may increase. In addition, the size of each stateincreases by the two boolean trend variables, for each colony in the MPP.

A Priori and A Posteriori Trend Computation. Trend variables provide an aposteriori detection of a stochastic trend, whereas trend formulae an a prioridetection. More precisely, if a transition from state s to state s′ increases (de-creases) the counts in a colony, the trend variable approach detects in state s′

the increasing (resp. decreasing) trend, whereas the trend formulae approach de-tects the trend in state s, i.e., prior to the transition. When deciding to analysean MPP using trends, one has to decide which type of detection of the trendbest suits the problem. Note also that the values of trend variables associatedwith a colony variable A are not updated during a transition if the value ofA is not changed by the transition. This notion of monotonicity corresponds,in our framework, to weak monotonicity, more precisely to very weakly increas-ing/decreasing trends.

We illustrate the difference between the increasing trend computed usingtrend variables and computed using the trend formulae for the MPP C1 fromFig. 6. Let C ′1 be the MPP resulting from adding trend variables to C1, asdepicted in Fig. 8. Now consider the temporal property φ =“eventually the value


Fig. 8. MPP C′1 with initial state s′0 = (4, t, t) obtained from C1 by adding trendvariables.

of i will increase”. In CSL with trends formulae this property is specified asP=?[F (θup ∧ (s = k))] for MPP C1 with θup either strictly increasing, weaklyincreasing or very weakly increasing trend and k ranging from 0 to 6. Thenφ′ = P=?[F (inc ∧ (s′ = k))] is the property corresponding to φ we want to checkfor C ′1. The results of model checking φ and φ′ in PRISM are given in Table 2.Notice that in C ′1 an increasing trend is found in the initial state only because thevariables inc and dec are set initially to true. Otherwise, an increasing trend isdetected in C ′1 in state s′2 because the transition from s′1 to s′2 increased the valueof i: the conclusion that the trend is increasing in state s′2 was established whenthe transition to be triggered was already chosen. Therefore we call this analysisa posteriori. The same reasoning can be applied to the increasing trend in states′3. The detection of state s2 as a state with increasing trend in C1 is performed apriori any possible transition and this trend corresponds to the increasing trenddetected in state s′3 of C ′1. With trend formulae we detect the states with thehighest probability of moving to a state where the value of i is increased. Thecorresponding state of the increasing trend of s′2 in C ′1 is s1 in C1 having a veryweakly increasing trend. The strictly increasing and weakly increasing trendsrequire that the probability of i to increase in state s1 is strictly greater than theprobability to decrease, which is not the case because Pi↑(s1) = Pi↓(s1) = 0.5.The very weakly increasing trend is detected in states s3 and s6 where, becauseof the cycle, the value of i remains unchanged.

The trend formulae approach permits expressing different concepts that can-not be expressed with trend variables such as several types of monotonicity or,for instance, the following property for C1: “What is the probability to eventuallyreach a state where i will most probably increase and in the next state will mostprobably decrease”. By model checking C1 |= P=?[F ((i ↑=) ∧ P>0[X (i⇓)])] weobtain probability 1 as the probability of taking the path starting in the initialstate that reaches the state s1 where i = 3 has the highest chances of increasingand in the next state to decrease strictly. But if we consider the CSL formula


Table 2. Comparison of checking trend formulae and trend variable properties in C1

and C′1

Model checking CSL formula k

0 1 2 3 4 5 6

C1 |= P=?[F ((i⇑) ∧ (s = k))] 0 0 0.5 0 0 0 0

C1 |= P=?[F ((i↑) ∧ (s = k))] 0 0 0.5 0 0 0 0

C1 |= P=?[F ((i↑=) ∧ (s = k))] 0 1 0.5 0.5 0 0 0.5

C′1 |= P=?[F (inc ∧ (s′ = k))] 1 0 0.5 0.5 0 0 0

with trend variables, C ′1 |= P=?[F (inc ∧ P>0[X dec])], the result is probability0 because it detects the increasing trend in state s′2, where i = 4 and from thenext state the trend is only increasing.

6 Trend-based Analysis of the AKAP System

In this section we apply trend formulae in the analysis of AKAP system. Weformalise in CSL the causality and fluctuation properties and model check themin PRISM. Stochastic trend formulae are essential for expressing these proper-ties. The key question is which trends best encode more X and less X for X acolony. Consider the statement more X. In order to express an increase in theconcentration of X, we rule out decreasing concentrations but consider transi-tions that do not change the concentration. Therefore the trend we choose toencode more X is the weakly increasing trend X ↑. The same reasoning appliesto encoding less X by X ↓.

6.1 Causality Relation

A causality relation between two events can be formalised as a temporal queryusing the necessarily preceded or requirement pattern [29]. This pattern rep-resents an ordering relation between two events, the occurrence of the latterbeing conditioned by the occurrence of the former: a state φ is reachable andis necessarily preceded all the time by a state ϕ. The associated CTL formula isEFφ ∧ (AG((¬ϕ) ⇒ AG(¬φ))), where A and E are temporal operators quan-tifying universally and existentially over paths respectively.

Consider now the causal relation stated in Section 3 for the AKAP model:

more pPDE8A1⇒ less cAMP and less active PKA

The two CSL state formulae encoding the two sides of the implication above are:

ϕ1 , pPDE8A1↑φ1 , (cAMP↓) ∧ (active PKA↓)


where the concentration of active PKA is given by the sum of concentrations of allscaffold combinations with 1 in the first position: S10, S11, S100, S110, S101 andS111. Employing basic proposition equivalences, we translate the requirementpattern for the cause ϕ1 and effect φ1 into CSL to obtain the following formulawhich was checked as true for our PRISM model:

P>0[Fφ1] ∧ P≥1[G((¬ϕ1)⇒ P≤0[Gφ1]))]

We can express a tighter causality relation between increasing concentrationlevels of pPDE8A1 (ϕ2 , pPDE8A1 ↑) and decreasing levels of cAMP (φ2 ,cAMP↓) using the following formula checked as true for our PRISM model:

P≥1[F ((¬ϕ2 ∧ ¬φ2) U (P≥1[(ϕ2 ∧ ¬φ2) UP>0[Xφ2]]))]

This formula stands for “more pPDE8A1 ⇒ less cAMP” in the notation intro-duced in Sect. 3 and it states that there is a time interval where the trendof pPDE8A1 is not decreasing and the trend of cAMP is not decreasing un-til the trend of pPDE8A1 starts increasing and soon after the trend of cAMPstarts decreasing. A similar CSL formula can be employed in order to show that“less pPDE8A1⇒ more cAMP” and “less cAMP⇒ less active PKA”.

6.2 Pulsating Behaviour

An oscillating behaviour of a variable assumes a fluctuation of the value ofthe variable around a given value k. Oscillation and its expression as temporalformulae in CTL and PCTL have been studied in [10] and informally describedas always in the future, the variable x departs from and reaches the values kinfinitely often. The corresponding CTL formula is AG(((x = k) ⇒ EF(x 6=k)) ∧ ((x 6= k) ⇒ EF(x = k))). In the context of BIOCHAM [9], a weakerform of oscillation properties expressed in CTL is used with the symbolic modelchecker NuSMV; the oscillating behaviour is approximated by the necessarybut not sufficient formula EG((EF¬ϕ) ∧ (EFϕ)) expressing that there existsa path where at all time points whenever ϕ is true it becomes eventually false,and whenever it is false it becomes eventually true.

We are interested in pulsating behaviour, i.e. no fixed k. We therefore con-sider oscillations (around 0) of the values of some variables. We refer to thisapproximate oscillating behaviour as pulsation. The CSL formulae describingpulsations of cAMP, active PKA and pPDE8A1 are the following:

P≥1[G(((cAMP↑)⇒ P>0[F(cAMP↓)]) ∧ ((cAMP↓)⇒ P>0[F(cAMP↑)]))]P≥1[G(((active PKA↑)⇒ P>0[F(active PKA↓)])∧

((active PKA↑)⇒ P>0[F(active PKA↓)]))]P≥1[G(((pPDE8A1↑)⇒ P>0[F(pPDE8A1↓)]) ∧ ((pPDE8A1↓)⇒ P>0[F(pPDE8A1↑)]))]

and they were all checked as true for our model using PRISM.We can also prove that the presence of a synchronised pulsation with pPDE8A1

showing a very weakly increasing (decreasing) trend at the same time as cAMP


and active PKA follow a very weakly decreasing (increasing) trend. Consider thefollowing two state formulae:

φ3 , (pPDE8A1↑=) ∧ (cAMP↓=) ∧ (active PKA↓=)

φ4 , (pPDE8A1↓=) ∧ (cAMP↑=) ∧ (active PKA↑=)

The following formula expressing a synchronised pulsation was checked as truefor our model using PRISM:

P≥1[G((φ3 ⇒ P>0[Fφ4]) ∧ (φ4 ⇒ P>0[Fφ3]))]

We remark that using weakly monotonic trends in formulae φ3 and φ4, theabove formula would return false. Hence weakly monotonic trends are too strongto show the synchronised pulsation, whereas the very weakly trends validate it.The reason is that the pulsations take place modulo a very small time delay, whenthe probability of increasing concentrations may be equal to the probabilityof decreasing concentration of a species. Therefore the three species (cAMP,active PKA and pPDE8A1) do pulsate in a synchronised way, but only when weconsider weak monotonicity.

Finally, we note that we have not used any timed operators, i.e. the boundeduntil operator, in this case study. This is because the system is still under in-vestigation and currently we have only semi-quantitative information. It wastherefore more relevant to consider trends within the context of unbounded tem-poral operators. In other applications, where rate information is more precise,time-bounded operators would be more relevant.

7 Conclusions and Future Work

We have introduced stochastic trend formulae for characterising the probabilityof increasing/decreasing colonies in MPP models. The probabilities are forward-looking, based on behaviour that will occur in the future. We defined a set ofstochastic trend formulae and showed how to derive several formulae encapsu-lating useful forms of monotonicity. We extended the set of state formulae ofCSL with trends formulae and we defined an encoding in the PRISM languageusing the PRISM formula construct, which means that there are no additionalvariables in the underlying state space. We compared stochastic trend formulaewith stochastic trend variables, and showed the former is more tractable withrespect to the state space size and the size of the states. We note that while wefocus on continuous time here, similar results are easily obtained for the discretetime case.

After illustration with several small examples, stochastic trends were appliedto the analysis of causality relations and pulsating behaviour in a significantbiochemical signalling case study: the AKAP mediated crosstalk between thecAMP and Raf-1/ERK/MEK pathways. We believe this to be the first formalmodel of this system. We were able, with the use of trend formulae, to showcausality and pulsations predicted by life scientists and observed in laboratoryexperiments.


Future work includes investigating how stochastic trends (an abstraction)over different combinations of colonies affects various relations (e.g. simulation)between MPPs.

Acknowledgements We thank George Baillie, Kim Brown and Walter Kolch fromthe Faculty of Biomedical & Life Science, University of Glasgow, for discussions,guidance and insight into the AKAP scaffold. We also thank the anonymousreviewers of this paper for their insightful comments on the work. This work wassupported by the SIGNAL project, funded by the UK Engineering and PhysicalScience Research Council (EPSRC) under grant number EP/E031439/1.

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